In this talk we will describe the behaviors of flat surfaces and geodesics on hyperbolic surfaces, as their genera tend to infinity. We first discuss enumerative results that count the number of such surfaces or geodesics (which can be viewed as volumes of particular moduli spaces) in the large genus limit, and then we will explain how a randomly sampled such object looks. The large genus limits of these counts will rely on asymptotic results on the behaviors of these intersection numbers at high genus, which might be of independent interest.